Question

1. which one doesnt belong? each set of numbers represents the side measures of a triangle. identify the set that does not belong with the other three. explain your reasoning. 3 - 4 - 5 12 - 35 - 37 3 - 5 - 7 6 - 8 - 10

Explanation:

Step1: Check if the sets satisfy the Pythagorean theorem (for right - angled triangles)

For \(3 - 4 - 5\):
\(3^{2}+4^{2}=9 + 16=25=5^{2}\)
For \(12 - 35 - 37\):
\(12^{2}+35^{2}=144+1225 = 1369=37^{2}\)
For \(6 - 8 - 10\):
\(6^{2}+8^{2}=36 + 64=100=10^{2}\)
For \(3 - 5 - 7\):
\(3^{2}+5^{2}=9+25 = 34
eq7^{2}=49\)

Step2: Check the triangle inequality theorem (for any triangle)

The triangle inequality theorem states that for a triangle with side lengths \(a\), \(b\), and \(c\), \(a + b>c\), \(a + c>b\), and \(b + c>a\)
For \(3 - 5 - 7\):
\(3+5=8>7\), \(3 + 7=10>5\), \(5+7=12>3\) (it is a valid triangle, but not a right - angled triangle)

Answer:

The set \(3 - 5 - 7\) doesn't belong. The sets \(3 - 4 - 5\), \(12 - 35 - 37\), and \(6 - 8 - 10\) are Pythagorean triples (they satisfy \(a^{2}+b^{2}=c^{2}\) and represent the side lengths of right - angled triangles), while \(3 - 5 - 7\) is just a valid triangle (by the triangle inequality theorem) but not a right - angled triangle (since \(3^{2}+5^{2}
eq7^{2}\)).